If a negative number is raised to an even exponent, what's the sign of the answer? How do you know? d. If a negative number is raised to an odd exponent, what's the sign of the answer? How do you know?
negative times negative is positive
... even exponent means pairs of negative numbers
... always positive result
odd exponent means one extra negative after the pairs are multiplied
... negative result
When a negative number is raised to an even exponent, the answer will be positive. This can be explained by the fact that when you multiply two negative numbers, the result is always positive. So, when a negative number is multiplied by itself an even number of times, the negatives cancel out and the result becomes positive.
On the other hand, when a negative number is raised to an odd exponent, the answer will be negative. This can be understood by considering that when you multiply two negative numbers, the result is positive. So, when a negative number is multiplied by itself an odd number of times, one negative sign remains, resulting in a negative number as the answer.
To understand the signs of the answer when a negative number is raised to an even or odd exponent, we can rely on a general rule in mathematics.
a. If a negative number is raised to an even exponent, the answer is positive.
To explain this, let's take a negative number, such as -2, and raise it to an even exponent, like 4 (-2^4).
To compute this, we multiply the base (-2) by itself repeatedly for the number of times indicated by the exponent (4).
(-2) * (-2) * (-2) * (-2) = 16
As you can see, even though we started with a negative number, the resulting answer is positive. This holds true for any negative number raised to an even exponent.
One way to understand why this happens is by considering the multiplication of negative numbers. When you multiply two negative numbers, the result is positive. So, as we multiply the negative number by itself repeatedly, the negative signs cancel each other out, resulting in a positive value.
b. If a negative number is raised to an odd exponent, the answer is negative.
Taking the same example as before, with the base -2 and an odd exponent of 3 (-2^3), we can calculate it as follows:
(-2) * (-2) * (-2) = -8
In this case, the answer is negative. This holds true for any negative number raised to an odd exponent.
To understand why this happens, consider that when we multiply two negative numbers together, the result is positive. So, when we multiply the negative base by itself repeatedly, the negative signs cancel out partially, leaving one remaining negative sign. Hence, the answer is negative.
In summary, when a negative number is raised to an even exponent, the answer is always positive, and when a negative number is raised to an odd exponent, the answer is always negative. The underlying reasoning lies in the concept of multiplying negative numbers and how the negative signs interact with each other.